A ratio limit theorem for (sub) Markov chains on {1,2, …} with bounded jumps

Abstract

We consider positive matricesQ,indexed by {1,2, …}. Assume that there exists a constant 1L <∞ and sequencesu1< u2< · ··andd1d2< · ··such thatQ(i, j) = 0 wheneveri<ur< ur+L < jori > dr+ L > dr> jfor somer. IfQsatisfies some additional uniform irreducibility and aperiodicity assumptions, then fors> 0,Qhas at most one positives-harmonic function and at most ones-invariant measureµ.We use this result to show that ifQis also substochastic, then it has the strong ratio limit property, that isfor a suitableRand someR–1-harmonic functionfandR–1-invariant measureµ.Under additional conditionsµcan be taken as a probability measure on {1,2, …} andexists. An example shows that this limit may fail to exist ifQdoes not satisfy the restrictions imposed above, even thoughQmay have a minimal normalized quasi-stationary distribution (i.e. a probability measureµfor whichR–1µ = µQ).The results have an immediate interpretation for Markov chains on {0,1,2, …} with 0 as an absorbing state. They give ratio limit theorems for such a chain, conditioned on not yet being absorbed at 0 by timen.